Theorem sbabel 2016

Description: Theorem to move a substitution in and out of a class abstraction. (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)

Hypothesis

Ref	Expression
sbabel.1	|- (w e. A -> A.x w e. A)

Assertion

Ref	Expression
sbabel	|- ([y / x]{z | ph} e. A <-> {z | [y / x]ph} e. A)

Distinct variable groups:   w,A   x,z   y,z

Proof of Theorem sbabel

Step	Hyp	Ref	Expression
1		sbex 1739	. . 3 |- ([y / x]E.v(v = {z | ph} /\ v e. A) <-> E.v[y / x](v = {z | ph} /\ v e. A))
2		sban 1607	. . . . 5 |- ([y / x](v = {z | ph} /\ v e. A) <-> ([y / x]v = {z | ph} /\ [y / x]v e. A))
3		sbal 1738	. . . . . . . 8 |- ([y / x]A.z(z e. v <-> ph) <-> A.z[y / x](z e. v <-> ph))
4		ax-17 1317	. . . . . . . . . . 11 |- (z e. v -> A.x z e. v)
5	4	sbf 1551	. . . . . . . . . 10 |- ([y / x]z e. v <-> z e. v)
6	5	sbrbis 1611	. . . . . . . . 9 |- ([y / x](z e. v <-> ph) <-> (z e. v <-> [y / x]ph))
7	6	albii 1346	. . . . . . . 8 |- (A.z[y / x](z e. v <-> ph) <-> A.z(z e. v <-> [y / x]ph))
8	3, 7	bitri 190	. . . . . . 7 |- ([y / x]A.z(z e. v <-> ph) <-> A.z(z e. v <-> [y / x]ph))
9		abeq2 1999	. . . . . . . 8 |- (v = {z | ph} <-> A.z(z e. v <-> ph))
10	9	sbbii 1538	. . . . . . 7 |- ([y / x]v = {z | ph} <-> [y / x]A.z(z e. v <-> ph))
11		abeq2 1999	. . . . . . 7 |- (v = {z | [y / x]ph} <-> A.z(z e. v <-> [y / x]ph))
12	8, 10, 11	3bitr4i 200	. . . . . 6 |- ([y / x]v = {z | ph} <-> v = {z | [y / x]ph})
13		sbabel.1	. . . . . . . 8 |- (w e. A -> A.x w e. A)
14	13	hblem 1993	. . . . . . 7 |- (v e. A -> A.x v e. A)
15	14	sbf 1551	. . . . . 6 |- ([y / x]v e. A <-> v e. A)
16	12, 15	anbi12i 540	. . . . 5 |- (([y / x]v = {z | ph} /\ [y / x]v e. A) <-> (v = {z | [y / x]ph} /\ v e. A))
17	2, 16	bitri 190	. . . 4 |- ([y / x](v = {z | ph} /\ v e. A) <-> (v = {z | [y / x]ph} /\ v e. A))
18	17	exbii 1398	. . 3 |- (E.v[y / x](v = {z | ph} /\ v e. A) <-> E.v(v = {z | [y / x]ph} /\ v e. A))
19	1, 18	bitri 190	. 2 |- ([y / x]E.v(v = {z | ph} /\ v e. A) <-> E.v(v = {z | [y / x]ph} /\ v e. A))
20		df-clel 1880	. . 3 |- ({z | ph} e. A <-> E.v(v = {z | ph} /\ v e. A))
21	20	sbbii 1538	. 2 |- ([y / x]{z | ph} e. A <-> [y / x]E.v(v = {z | ph} /\ v e. A))
22		df-clel 1880	. 2 |- ({z | [y / x]ph} e. A <-> E.v(v = {z | [y / x]ph} /\ v e. A))
23	19, 21, 22	3bitr4i 200	1 |- ([y / x]{z | ph} e. A <-> {z | [y / x]ph} e. A)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  {cab 1871

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880

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